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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Graeffe's method ： ウィキペディア英語版 Graeffe's method In mathematics, Graeffe's method or Dandelin–Graeffe method is an algorithm for finding all of the roots of a polynomial. It was developed independently by Germinal Pierre Dandelin in 1826 and Karl Heinrich Gräffe in 1837. Lobachevsky in 1834 also discovered the principal idea of the method.〔Alston Scott Householder: ''Dandelin, Lobačevskiǐ, or Graeffe?'', Amer. Math. Monthly, 66 (1959), pp. 464–466 ((on JSTOR) )〕 The method separates the roots of a polynomial by squaring them repeatedly. This squaring of the roots is done implicitly, that is, only working on the coefficients of the polynomial. Finally, Viète's formulas are used in order to approximate the roots. ==Dandelin–Graeffe iteration== Let be a polynomial of degree :$p(x)\; =\; (x-x\_1)\backslash cdots(x-x\_n).$ Then :$p(-x)\; =\; (-1)^n\; (x+x\_1)\backslash cdots(x+x\_n).$ Let be the polynomial which has the squares $x\_1^2,\; \backslash cdots,\; x\_n^2$ as its roots, :$q(x)=\; \backslash left\; (x-x\_1^2\; \backslash right\; )\backslash cdots\; \backslash left\; (x-x\_n^2\; \backslash right\; ).$ Then we can write: :$\backslash begin$ q(x^2) & = \left (x^2-x_1^2 \right )\cdots \left (x^2-x_n^2 \right ) \\ & = (x-x_1)(x+x_1) \cdots (x-x_n) (x+x_n) \\ & = \left \ \times \left \ \\ & = p(x) \times \left \ \\ & = p(x) \times \left \ \\ & = (-1)^n p(x) p(-x) \end can now be computed by algebraic operations on the coefficients of the polynomial alone. Let: :$\backslash begin$ p(x) &= x^n+a_1x^+\cdots+a_x+a_n \\ q(x) &= x^n+b_1x^+\cdots+b_x+b_n \end then the coefficients are related by :$b\_k=(-1)^k\; a\_k^2\; +\; 2\backslash sum\_^(-1)^j\backslash ,a\_ja\_,\; \backslash qquad\; a\_0=b\_0=1.$ Graeffe observed that if one separates into its odd and even parts: :$p(x)=p\_e\; \backslash left\; (x^2\; \backslash right\; )+x\; p\_o\backslash left\; (x^2\; \backslash right\; ),$ then one obtains a simplified algebraic expression for : :$q(x)=(-1)^n\; \backslash left\; (p\_e(x)^2-x\; p\_o(x)^2\; \backslash right\; ).$ This expression involves the squaring of two polynomials of only half the degree, and is therefore used in most implementations of the method. Iterating this procedure several times separates the roots with respect to their magnitudes. Repeating ''k'' times gives a polynomial of degree : :$q^k(y)\; =\; y^n\; +\; \_1\backslash ,y^\; +\; \backslash cdots\; +\; \_\backslash ,y\; +\; \_n\; \backslash ,$ with roots :$y\_1=x\_1^,\backslash ,y\_2=x\_2^,\backslash ,\backslash dots,\backslash ,y\_n=x\_n^.$ If the magnitudes of the roots of the original polynomial were separated by some factor $\backslash rho>1$, that is, $|x\_k|\backslash ge\backslash rho\; |x\_|$, then the roots of the ''k''-th iterate are separated by a fast growing factor :$\backslash rho^\backslash ge\; 1+2^k(\backslash rho-1)$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Graeffe's method」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.